We would like to acknowledge the financial support received from
Shaqra University. With our sincere thanks and appreciation to Professor
Smarandache for his support and his comments.

1 Introduction

Many scientists wish to find appropriate solutions to some
mathematical problems that cannot be solved by traditional methods.
These problems lie in the fact that traditional methods cannot solve the
problems of uncertainly in economy, engineering, medicine, problems of
decision-making and others. There have been a great amount of research
and applications in the literature concerning some special tools like
probability theory, fuzzy set theory [13], rough set theory [19], vague
set theory [18], intuitionistic fuzzy set theory [10, 12] and interval
mathematics [11, 14].

Since Zadeh published his classical paper almost fifty years ago,
fuzzy set theory has received more and more attention from researchers
in a wide range of scientific areas, especially in the past few years.

The difference between a binary set and a fuzzy set is that in a
"normal" set every element is either a member or a non-member
of the set; it either has to be A or not A.

In a fuzzy set, an element can be a member of a set to some degree
and at the same time a non-member of the same set to some degree. In
classical set theory, the membership of elements in a set is assessed in
binary terms: according to a bivalent condition, an element either
belongs or does not belong to the set.

By contrast, fuzzy set theory permits the gradual assessment of the
membership of elements in a set; this is described with the aid of a
membership function valued in the closed unit interval [0, 1].

Fuzzy sets generalise classical sets, since the indicator functions
of classical sets are special cases of the membership functions of fuzzy
sets, if the later only take values 0 or 1. Therefore, a fuzzy set A in
an universe of discourse X is a function A:X [right arrow] [0,1], and
usually this function is referred to as the membership function and
denoted by [[mu].sub.A(x)].

The theory of vague sets was first proposed by Gau and Buehrer [18]
as an extension of fuzzy set theory and vague sets are regarded as a
special case of context-dependent fuzzy sets.

A vague set is defined by a truth-membership function tv and a
false-membership function [f.sub.v], where [t.sub.v](x) is a lower bound
on the grade of membership of x derived from the evidence for x, and
[f.sub.v](x) is a lower bound on the negation of x derived from the
evidence against x. The values of [t.sub.v](x) and [f.sub.v](x) are both
defined on the closed interval [0, l] with each point in a basic set X,
where [t.sub.v](x) + [f.sub.v](x) [less than or equal to] 1.

For more information, see [1, 2, 3, 7, 15, 16, 19].

In 1995, Smarandache talked for the first time about neutrosophy,
and in 1999 and 2005 [4, 6] defined the neutrosophic set theory, one of
the most important new mathematical tools for handling problems
involving imprecise, indeterminacy, and inconsistent data.

In this paper, we define the concept of a neutrosophic vague set as
a combination of neutrosophic set and vague set. We also define and
study the operations and properties of neutrosophic vague set and give
examples.

2 Preliminaries

In this section, we recall some basic notions in vague set theory
and neutrosophic set theory. Gau and Buehrer have introduced the
following definitions concerning its operations, which will be useful to
understand the subsequent discussion.

Definition 2.3 ([18]). Let A be a vague set of the universe U. If
[for all][u.sub.i] [member of] U, [t.sub.A]([u.sub.t]) = 1 and
[f.sub.A]([u.sub.t]) = 0, then A is called a unit vague set, where 1
[less than or equal to] i [less than or equal to] n. If [for
all][u.sub.i] [member of] U, [t.sub.A] ([u.sub.i]) = 0 and
[f.sub.A]([u.sub.i]) = 1, then A is called a zero vague set, where 1
[less than or equal to] i [less than or equal to] n.

Definition 2.4 ([18]). The complement of a vague set A is denoted
by [A.sup.c] and is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII].

Definition 2.5 ([18]). Let A and B be two vague sets of the
universe U. If [for all][u.sub.i] [member of] U, [[t.sub.A] ([u.sub.i]),
l - [f.sub.A]([u.sub.i])] = [[t.sub.B] ([u.sub.i]), l -
[f.sub.B]([u.sub.i])]. then the vague set A and B are called equal,
where 1 [less than or equal to] i [less than or equal to] n.

Definition 2.6 ([18]). Let A and B be two vague sets of the
universe U. If [for all][u.sub.i] [member of] U, [[t.sub.A] ([u.sub.i])
[less than or equal to] [[t.sub.B] ([u.sub.i]) and 1 - [[f.sub.A]
([u.sub.i]) [less than or equal to] [[f.sub.B] ([u.sub.i]), then the
vague set A are included by B, denoted by A [subset or equal to] B,
where 1 [less than or equal to] i [less than or equal to] n.

Definition 2.7 ([18]). The union of two vague sets A and B is a
vague set C, written as C = A [union] B, whose truth-membership and
false-membership functions are related to those of A and B by

Definition 2.8 ([18]). The intersection of two vague sets A and B
is a vague set C, written as C = A[intersection]B, whose
truth-membership and false-membership functions are related to those of
A and B by

Smarandache explained his concept as it follows: "For example,
neutrosophic logic is a generalization of the fuzzy logic. In
neutrosophic logic a proposition is T [equivalent to] true, I
[equivalent to] indeterminate, and F [equivalent to] false. For example,
let's analyze the following proposition: Pakistan will win against
India in the next soccer game. This proposition can be (0.6,0.3,0.1),
which means that there is a possibility of 60% [equivalent to] that
Pakistan wins, 30% [equivalent to] that Pakistan has a tie game, and 10%
[equivalent to] that Pakistan looses in the next game vs. India."

Now we give a brief overview of concepts of neutrosophic set
defined in [8, 5, 17]. Let [S.sub.1] and [S.sub.2] be two real standard
or non-standard subsets, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.10 (Containment) A neutrosophic set A is contained in
the other neutrosophic set B, A [subset or equal to] B, if and only if

Definition 2.12 (Intersection) The intersection of two neutrosophic
sets A and B is a neutrosophic set C, written as C = A [intersection] B,
whose truth-membership, indeterminacy-membership and falsity-membership
functions are related to those of A and B by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 2.11 (Union) The union of two neutrosophic sets A and B
is a neutrosophic set C written as C = A [union] B, whose
truth-membership, indeterminacy-membership and falsity-membership
functions are related to those of A and B by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Neutrosophic Vague Set

A vague set over U is characterized by a truth-membership function
[t.sub.v] and a false-membership function [f.sub.v], [t.sub.v] : U
[right arrow] [0,1] and [f.sub.v] : U [right arrow] [0,1] respectively
where [t.sub.v] ([u.sub.i]) is a lower bound on the grade of membership
of [u.sub.i] which is derived from the evidence for [u.sub.i], [f.sub.v]
([u.sub.i]) is a lower bound on the negation of [u.sub.i] derived from
the evidence against [u.sub.i] and [t.sub.v] ([u.sub.i]) + [f.sub.v]
([u.sub.i]) [less than or equal to] 1. The grade of membership of
[u.sub.i] in the vague set is bounded to a subinterval [[t.sub.v]
([u.sub.i]), 1 - [f.sub.v] ([u.sub.i])] of [0,1]. The vague value
[[t.sub.v] ([u.sub.i]), 1 - [f.sub.v] ([u.sub.i])] indicates that the
exact grade of membership [[mu].sub.v] ([u.sub.i]) of [u.sub.i] maybe
unknown, but it is bounded by [t.sub.v] ([u.sub.i]) [less than or equal
to] [f.sub.v] ([u.sub.i]) where [t.sub.v] ([u.sub.i]) [less than or
equal to]1 . Let U be a space of points (objects), with a generic
element in U denoted by u. A neutrosophic sets (N-sets) A in U is
characterized by a truth-membership function [T.sub.A], an
indeterminacy-membership function [I.sub.A] and a falsity-membership
function [F.sub.A]. [T.sub.A] (u); [I.sub.A] (u) and [F.sub.A] (u) are
real standard or nonstandard subsets of [0, 1]. It can be written as:

Also, vague logic is a generalization of the fuzzy logic where a
proposition is T [equivalent to] true and F [equivalent to] false, such
that: [t.sub.v]([u.sub.i]) + [f.sub.v] ([u.sub.i]) [less than or equal
to] 1, he exact grade of membership [[mu].sub.v] ([u.sub.i]) of
[u.sub.i] maybe unknown, but it is bounded by

For example, let's analyze the Smarandache's proposition
using our new concept: Pakistan will win against India in the next
soccer game. This proposition can be as it follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which means that there is possibility of 60% to 90% [equivalent to]
that Pakistan wins, 30% to 40% [equivalent to] that Pakistan has a tie
game, and 40% to 60% [equivalent to] that Pakistan looses in the next
game vs. India.

Example 3.1 Let u = {[u.sub.1], [u.sub.2], [u.sub.3]} be a set of
universe we define the NVS [A.sub.NV] as follows:

then [[PSI].sub.NV] is called a unit NVS, where 1 [less than or
equal to] i [less than or equal to] n.

Let [[PHI].sub.NV] be a NVS of the universe U where [for
all][u.sub.i] [member of] U,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then [[PHI].sub.NV] is called a zero NVS, where 1 [less than or
equal to] i [less than or equal to] n.

Definition 3.3 The complement of a NVS [A.sub.NV] is denoted by
[A.sup.c] and is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Example 3.2 Considering Example 3.1, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Definition 3.5 Let [A.sub.NV] and [B.sub.NV] be two NVSs of the
universe U. If [for all][u.sub.i] [member of] U,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then the NVS [A.sub.NV] and [B.sub.NV] are called equal, where 1
[less than or equal to] i [less than or equal to] n.

Definition 3.6 Let [A.sub.NV] and [B.sub.NV] be two NVSs of the
universe U. If [for all][u.sub.i] [member of] U,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then the NVS [A.sub.NV] are included by [B.sub.NV], denoted by
[A.sub.NV] [subset or equal to][B.sub.NV], where 1 [less than or equal
to] i [less than or equal to] n.

Definition 3.7 The union of two NVSs [A.sub.NV] and [B.sub.NV] is a
NVS [C.sub.NV], written as [C.sub.NV] = [A.sub.NV] [union] [B.sub.NV],
whose truth-membership, indeterminacy-membership and false-membership
functions are related to those of [A.sub.NV] and [B.sub.NV] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 3.8 The intersection of two NVSs [A.sub.NV] and
[B.sub.NV] is a NVS [H.sub.NV], written as [H.sub.NV] = [A.sub.NV]
[intersection] [B.sub.NV], whose truth-membership,
indeterminacy-membership and false-membership functions are related to
those of [A.sub.NV] and [B.sub.NV] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.3 Let u = {[u.sub.1], [u.sub.2], [u.sub.3]} be a set of
universe and let [A.sub.NV] and [B.sub.NV] define as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have [c.sub.NV] = [A.sub.NV] [union] [B.sub.NV] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, we have [H.sub.NV] = [A.sub.NV] [intersection] [B.sub.NV]
where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.1 Let P be the power set of all NVS defined in the
universe X. Then <P; [[union].sub.NV], [[intersection].sub.NV]> is
a distributive lattice.

Proof Let A, B, C be the arbitrary NVSs defined on X. It is easy to
verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4 Conclusion

In this paper, we have defined and studied the concept of a
neutrosophic vague set, as well as its properties, and its operations,
giving some examples.